Using elementary tansormations, find the inverse of each of the matrices, if it exists in

Engineering

Mathematics

Algebra of Matrices

Inverse of a Matrix

Question

Using elementary tansormations, find the inverse of each of the matrices, if it exists in $[\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}]$

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$A = [\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}]$

|A| = 2 – 1 = 1 ≠ 0

Hence, A^–1 exists.

Now, A = lA

[\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}] = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] A

[\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}] = [\begin{array}{cc} 1 & - 1 \\ 0 & 1 \end{array}] A

(R₁ → R₁ – R₂)

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 1 & - 1 \\ - 1 & 2 \end{array}] A

(R₂ → R₂ – R₁)

Hence,

A^{- 1} = [\begin{array}{cc} 1 & - 1 \\ - 1 & 2 \end{array}]